Biaccessibility in unicritical Julia sets(Complex Dynamics and its Related Topics)

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Biaccessibility in unicritical

Julia

sets

Mitsuhiko

Imada

Department

of

Mathematics, Tokyo

Institute

of Technology

E–mail: [email protected]

October

2,

2006

Abstract

We consider the dynamics of a degree $d\geq 2$ polynomial with only

one bounded critical point. Such a polynomial is often said to be a

unicritical polynomial. We can conjugate the polynomial to $f(z)=$

$z^{d}+c$ by an appropriate transformation. If $f$ has an irrationally

indifferent fixed point then the filled Julia set $K$ is connected. So we

can use external rays for $K$

.

We areinterested in how many rays land

at a

common

point for studying the topology of the Julia set. In fact,

points which are landing points of two or more rays are cut points of

the Julia set. Such points are said to be biaccessible. D. Schleicher

and S. Zakeri studied which points are biaccessible when $d=2$

.

We extend the result and consider when $d\geq 2$

.

1 Preliminaries

In this paper,

we

set $f(z)=z^{d}+c$ for

some

$d$ greater than

or

equal to 2.

Thus the bounded critical point is $0$

.

Recall that the filled Julia set of _$f$ is

$K^{\mathrm{d}}=^{\mathrm{e}\mathrm{f}}$

{

$z\in \mathbb{C}:\{f^{\mathrm{n}}(z)\}_{n\geq 0}$ is

bounded}

and the Julia set of $f$ is $J^{\mathrm{d}}=^{\mathrm{e}\mathrm{f}}\partial K$

.

Let $\tau_{j}(z)^{\mathrm{d}}=^{\mathrm{e}\mathrm{f}}e^{2\pi i}\dot{:}z(0\leq j\leq d-1)$ be

$\mathrm{a}\frac{j}{d}$-rotation. $f(\tau_{j}(z))=f(z)$ implies

$\tau_{j}(K)=K$ and thus $\tau_{j}(J)=J$

.

Let

us now

assume

that $K$

is

connected. Then there exists

a

unique

conformal

isomorphism

_th

: $\mathbb{C}\backslash \overline{\mathrm{D}}arrow \mathbb{C}\backslash K$ such that $\lim_{zarrow\infty}\frac{\psi(z)}{z}=1$

.

Moreover, the

following holds:

$f(\psi(z))=\psi(z^{d})$ $(\forall z\in \mathbb{C}\backslash \overline{\mathrm{D}})$. $(*)$

数理解析研究所講究録

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We say $R_{t}\mathrm{d}\mathrm{e}\mathrm{f}=\{\psi(re^{2\pi it}) : 1 <r\}$ is the external

ray with

angle $t\in \mathbb{R}/\mathbb{Z}$

.

Now $(*)$ implies $f(R_{t})=R_{dt}$

.

$\tau_{j}(K)=K$ implies $\tau_{j}\circ\psi=\psi\circ\tau_{j}$ and thus $\tau_{j}(R_{t})=R_{\iota+_{d}^{i}}$

. A

ray $R_{t}$ lands at $z$ if _{$\lim_{f\searrow 1}\psi(re^{2\pi it})=z$}

.

If two

or more

rays

land at $z$ then

we

say $z$ is biaccessible. By a thorem of F. and M. Riesz [Mi],

the point $z$ is

a

cut point in the Julia set, namely $J\backslash \{z\}$ is disconnected.

2 Main result

Lemma 1

Suppose that $K$

is

connected. Let $\alpha$ be

a

fixed

point of $f$

.

Suppose that $z$ is

a

biaccessible point such that $\alpha\not\in\{f^{n}(z)\}_{n\geq 0}$ and $0\not\in$

$\{f^{n}(z)\}_{n\geq 0}$

.

Then there exists two distinct

rays

$R_{t_{1}}$ and $R_{\ell_{2}}$ with

a

common

landing point $w$

,

such that $R_{t_{1}}\cup R_{t_{2}}\cup\{w\}$ separates

a

from $0$

.

Theorem 2 Let

a

be

an

indifferent fixed point of $f$

.

Let $z$ be

a

biaccessible

point. Then:

$\bullet$ in the parabolic

case,

$\alpha\in\{f^{n}(z)\}_{n\geq 0;}$ $\bullet$ in the Siegel case, $0\in\{f^{n}(z)\}_{n\geq 0;}$

$\bullet$ in the Cremer case,

a

$\in\{f^{n}(z)\}_{n\geq 0}$

or

$0\in\{f^{n}(z)\}_{n\geq 0}$

.

(If

or

has the smallcyclespropertythen$0\not\in\{f^{n}(z)\}_{n\geq 0}$ by [Ki, Th. 1.1].)

Remark 3 The proof of Theorem 2 is based

on

Lemma 1. We

can

show

Lemma 1 and Theorem 2 by arguments similar to those in [Za].

Theorem 4 ([Za] Th.5) Suppose that $f(z)=z^{2}+c$ _has

an

_irrationally

indifferent

fixed

point

a.

Let $z$ be

a

biaccessible

point.

Then.:

$\bullet$ in the Siegel case, $0\in\{f^{n}(z)\}_{n\geq 0;}$ $\bullet$ in the Cremer case, _{$\alpha\in\{f^{n}(z)\}_{n\geq 0}$}

.

References

[Ki] J. Kiwi. Non-accessible critical points of Cremer polynomials. Ergod.

Th.

&

Dynam. Sys.

20

(2000),

1391-1403.

[Mc]

C. McMullen. Complex

Dynamics and

Renormalization. Princeton

Uni-versity Press,

1994.

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[Mi] J. Milnor. Dynamics in One Complex Variable, 3rd edn. Princeton

Uni-versity Press,

2006.

[PM] R. P\’erez-Marco. Fixed points and circle maps. Acta Math.

179

(1997),

243-294.

[Po] C. Pommerenke. Boundary Behaviour

_{of Conformal}

Maps. Springer,

1992.

[SZ] D. Schleicher and S. Zakeri.

On

biaccessible points in the Julia set of

a

Cremer

quadratic polynomial. Proc. Amer. Math. 128 (1999),

933-937.

[Za] S. Zakeri. Biaccessibility in quadratic Julia sets. Ergod. Th. $\mathcal{B}$ Dynam.

Sys. 20 (2000),